A mysterious castaway claiming to have been lost at sea for 13 months is now safely back on land, but many questions remain about how he could have lived on his small boat for so long as it drifted across the Pacific Ocean.

The man calling himself Jose Salvador Alvarenga turned up in a heavily damaged boat on a remote coral atoll in the Marshall Islands, claiming that he had been living off fish and turtles he had caught and relying on rainwater, and sometimes his own urine, to drink.

Authorities are trying to determine the veracity of Alvarenga's story. The Mexican government issued a statement Monday confirming Alvarenga's identity and saying he was an El Salvador national who was living in Tonala in Chiapas state.

Alvarenga is from Garita Palmera in El Salvador, where CNN caught up with some members of his family. They hadn't heard from him in ages and thought he might be dead. He has a 12-year-old daughter there who doesn't remember her father.

"My heart would tell me that my son was not dead, but I wondered about it so often that I had started to lose faith," said Julia Alvarenga, his mother.

"God willing, my son is not dead. God willing, my son is alive. And we're going to see him again one day. I'm very happy after learning that he's alive and that we will have him back home soon," his father, Ricardo, said.

Alvarengawas found on sparsely populated Ebon Atoll, a 22-hour boat ride from the Marshall Islands capital of Majuro, on Thursday. The southernmost of the Marshall Islands' atolls, Ebon has only 2.2 square miles of land, one phone line and no Internet service. The government airplane that services the atoll was not working, so Alvarenga did not make it to Majuro until Monday morning.

Video from Majuro shows Alvarenga walking a gangplank from a government boat to a waiting ambulance. Waving to those gathered around the dock, he is supported by a medical assistant as he walks. From inside the ambulance, he gives a thumbs up before it drives away.

Alvarenga, who says he is 37, is now in a local hospital recovering from his ordeal, said Tom Armbruster, U.S. ambassador to the Marshall Islands.

"He's in much better shape than one would expect after such an ordeal," Armbruster said.

"I had just killed a bird to eat and saw some trees," he is quoted as saying.

"I cried, 'Oh, God.' I got to land and had a mountain of sleep. In the morning, I woke up and heard a rooster and saw chickens and saw a small house. I saw two native women screaming and yelling. I didn't have any clothes; I was only in my underwear, and they were ripped and torn," The Telegraph quotes Alvarenga as saying.

People on the island where he was found Thursday say the 26-foot fiberglass boat was in very bad condition, covered in barnacles and with the carcasses of several turtles littering the deck.

Alvarenga is said to have set off from the Mexican town of Costa Azul, according to Reynaldo Aguilar Martinez, undersecretary for the state of Chiapas. He claims to have left for what was supposed to be a one-day expedition to catch sharks on December 21, 2012.

Alvarenga said that he and a teenage companion were blown off-course by northerly winds and then caught in a storm, eventually losing use of their engines.

According to Anjenette Kattil of the Marshall Islands Ministry of Foreign Affairs, Alvarenga said that four weeks into their drift, he lost the young man because he refused to eat raw birds. There are no details on what Alvarenga did with the young man's body.

Alvarenga told the Telegraph his companion's death had him contemplating suicide.

"For four days, I wanted to kill myself. But I couldn't feel the desire; I didn't want to feel the pain. I couldn't do it," he is quoted as saying.

Kattil said Alvarenga worked for a company named Camaroneras de la Costa in Mexico. He has told authorities that he is a citizen of El Salvador but has lived in Mexico for the past 15 years and wishes to be repatriated back to Mexico.

Armbruster, the U.S. ambassador, said Alvarenga indicated that he had relatives living the United States and U.S. officials would attempt to locate them.

Government officials have been in contact with Mexico's ambassador to the Marshall Islands, who is based in the Philippines, concerning Alvarenga in hopes he can contact El Salvadoran authorities.

The Mexican Ministry of Foreign Affairs issued a statement saying it has sent personnel from its embassy in the Philippines "to learn directly about the case."

If Alvarenga's story proves true, the trip across the Pacific would have taken him across roughly 6,700 miles (10,800 kilometers) of open ocean before ending in the Marshall Islands, about halfway between Hawaii and Australia, in the northern Pacific.

Such an amazing journey isn't unheard-of in the small Pacific nation, as three Mexican fishermen made a similar drift voyage in 2006 that lasted nine months. Those men lived off fish they caught and rainwater, and they read the Bible for comfort.

Conditions in the Pacific make the timeline of Alvarenga's journey plausible, according to Judson Jones, a producer for CNN Weather.

Jones said the currents between Mexico and the Marshall Islands would have carried a boat about 27 miles (42 kilometers) a day. That would mean the journey would take about 208 days if the boat stayed in the current. But Jones said a meandering journey in and out of the currents was most likely, making a 13-month journey believable.